Measuring Segments and Angles

HONORS GEOMETRY

Postulates

RULER POSTULATE

The points of a line can be put into 1 to 1 correspondence with the real numbers so that the distance between any 2 points is the absolute value of the difference of the corresponding numbers.

A B

AB = |2 – 5| = 3

RULER POSTULATE

•Same measure•Notation: •Mark up the picture with the same marks

Congruent

AB CD A B C D

Let’s Practice

P T Q

Given: PT = 5x + 3 and TQ = 7x – 9Find: PT.

x = 6, PT = 33 Answers

Postulates

SEGMENT ADDITION POSTULATE

If 3 points A, B, and C are collinear and B is between A and C, then AB + BC = AC.

Let’s Practice

D S T

Given:DT = 60 and DS = 2x – 8 and ST = 3x – 12Find: x, DS, & ST.

x = 16, DS = 24, ST = 36 Answers

Let’s Practice

E F G

Given:EG = 100 and EF= 4x – 20 and FG = 2x + 30Find: x, EF, & FG.

x = 15, EF = 40, FG = 60 Answers

•A point that divides a segment into two congruent parts.•A line, a ray, or a segment can bisect another segment.

Midpoint

•Cut through at the midpoint•A line, a ray, or a segment can bisect another segment.

Bisect

•Cut into three equal parts•A line, a ray, or a segment can trisect another segment.

Trisect

Let’s PracticeA C B

GivenOC bisects AB, AC = 2x + 1and CB = 3x - 4Find: AC, CB, & AB.

O

x = 5, AC = 11, CB = 11, AB = 22 Answers

Postulates

PROTRACTOR POSTULATELet OA and OB be opposite rays in a plane. OA, OB, and all rays with endpoint O that can be drawn on one side of AB can be paired with the real numbers 0 to 180 so that OA is paired with 0 and OB is paired with 180. If OC is paired with x and OD is paired with y, then mCOD = |x – y|.

PROTRACTOR POSTULATE

O

A

B

CD

mCOD = |x – y| = |50 – 120| = 70

Types of Angles

Type Angle Range

Sketch

Acute 0 < x < 90

Right x = 90

Obtuse 90 < x < 180

Straight

x = 180

Active Inspire Protractor Demonstration

Postulates

ANGLE ADDITION POSTULATEIf point B is in the interior of AOC, then mAOB + mBOC = mAOC.

If AOC is a straight angle, then mAOB + mBOC = 180.

Let’s Practice

R

T

S

Given:mRST = 50 and mRSW = 125.Find:mTSW

W

mTSW = 75Answers

Let’s Practice

E

G

D

Given:DEF is a straight angle and mDEG = 145.Find:mGEF

F

mGEF = 35Answers

Angles and the Clock

Estimate the measure of the angle formed by the hands of a clock at:

6:004:405:20

Given:mMQV = 90 and mVQP = 35.Find:mMQP

Let’s PracticeP

M N

Q

V

mMQP = 125Answers

Given:mMVQ = 55Find:mQVP

Let’s PracticeP

M N

Q

V

mQVP = 125Answers

Judging by appearance, name two acute angles.

Let’s PracticeP

M N

Q

V

Judging by appearance, name two obtuse angles.

Given:mAOC = 7x – 2,mAOB = 2x + 8 and mBOC = 3x + 14.Find:x

Let’s Practice

C

O D

A B

x = 12Answers

Given:mAOB = 28 ,mBOC = 3x – 2 and mAOD = 6x.Find:x

Let’s Practice

C

O D

A B

x = 18Answers

•The segment joining the midpoint of a side to the opposite vertex.

Median of a Triangle

C is the midpoint of AB. D is the midpoint of AC. E is the midpoint of AD. F is the midpoint of ED. G is the midpoint of EF. H is the midpoint of DB.

If DC = 16, find GH.

Challenge

A BCDE FG H

1616

8 8

32

48

2424

44

2

GH = GF + FD + DHGH = 2 + 4 + 24GH = 30